MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { even(0()) -> true() , even(s(0())) -> false() , even(s(s(x))) -> even(x) , half(0()) -> 0() , half(s(s(x))) -> s(half(x)) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) , times(0(), y) -> 0() , times(s(x), y) -> if_times(even(s(x)), s(x), y) , if_times(true(), s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) , if_times(false(), s(x), y) -> plus(y, times(x, y)) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { even^#(0()) -> c_1() , even^#(s(0())) -> c_2() , even^#(s(s(x))) -> c_3(even^#(x)) , half^#(0()) -> c_4() , half^#(s(s(x))) -> c_5(half^#(x)) , plus^#(0(), y) -> c_6(y) , plus^#(s(x), y) -> c_7(plus^#(x, y)) , times^#(0(), y) -> c_8() , times^#(s(x), y) -> c_9(if_times^#(even(s(x)), s(x), y)) , if_times^#(true(), s(x), y) -> c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y))) , if_times^#(false(), s(x), y) -> c_11(plus^#(y, times(x, y))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { even^#(0()) -> c_1() , even^#(s(0())) -> c_2() , even^#(s(s(x))) -> c_3(even^#(x)) , half^#(0()) -> c_4() , half^#(s(s(x))) -> c_5(half^#(x)) , plus^#(0(), y) -> c_6(y) , plus^#(s(x), y) -> c_7(plus^#(x, y)) , times^#(0(), y) -> c_8() , times^#(s(x), y) -> c_9(if_times^#(even(s(x)), s(x), y)) , if_times^#(true(), s(x), y) -> c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y))) , if_times^#(false(), s(x), y) -> c_11(plus^#(y, times(x, y))) } Strict Trs: { even(0()) -> true() , even(s(0())) -> false() , even(s(s(x))) -> even(x) , half(0()) -> 0() , half(s(s(x))) -> s(half(x)) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) , times(0(), y) -> 0() , times(s(x), y) -> if_times(even(s(x)), s(x), y) , if_times(true(), s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) , if_times(false(), s(x), y) -> plus(y, times(x, y)) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,4,8} by applications of Pre({1,2,4,8}) = {3,5,6}. Here rules are labeled as follows: DPs: { 1: even^#(0()) -> c_1() , 2: even^#(s(0())) -> c_2() , 3: even^#(s(s(x))) -> c_3(even^#(x)) , 4: half^#(0()) -> c_4() , 5: half^#(s(s(x))) -> c_5(half^#(x)) , 6: plus^#(0(), y) -> c_6(y) , 7: plus^#(s(x), y) -> c_7(plus^#(x, y)) , 8: times^#(0(), y) -> c_8() , 9: times^#(s(x), y) -> c_9(if_times^#(even(s(x)), s(x), y)) , 10: if_times^#(true(), s(x), y) -> c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y))) , 11: if_times^#(false(), s(x), y) -> c_11(plus^#(y, times(x, y))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { even^#(s(s(x))) -> c_3(even^#(x)) , half^#(s(s(x))) -> c_5(half^#(x)) , plus^#(0(), y) -> c_6(y) , plus^#(s(x), y) -> c_7(plus^#(x, y)) , times^#(s(x), y) -> c_9(if_times^#(even(s(x)), s(x), y)) , if_times^#(true(), s(x), y) -> c_10(plus^#(times(half(s(x)), y), times(half(s(x)), y))) , if_times^#(false(), s(x), y) -> c_11(plus^#(y, times(x, y))) } Strict Trs: { even(0()) -> true() , even(s(0())) -> false() , even(s(s(x))) -> even(x) , half(0()) -> 0() , half(s(s(x))) -> s(half(x)) , plus(0(), y) -> y , plus(s(x), y) -> s(plus(x, y)) , times(0(), y) -> 0() , times(s(x), y) -> if_times(even(s(x)), s(x), y) , if_times(true(), s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) , if_times(false(), s(x), y) -> plus(y, times(x, y)) } Weak DPs: { even^#(0()) -> c_1() , even^#(s(0())) -> c_2() , half^#(0()) -> c_4() , times^#(0(), y) -> c_8() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..